Infinite-dimensional diffusions as limits of random walks on partitions

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dc.creatorBorodin, Alexei
dc.creatorOlshanski, Grigori
dc.date2007-06-07
dc.date2008-02-24
dc.date.accessioned2026-07-07T12:41:07Z
dc.date.available2026-07-07T12:41:07Z
dc.descriptionThe present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.
dc.descriptionAMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Fields
dc.identifierhttps://arxiv.org/abs/0706.1034
dc.identifierhttp://arxiv.org/abs/0706.1034
dc.identifierProb. Theor. Rel. Fields 144 (2009), no. 1, 281-318
dc.identifierdoi:10.1007/s00440-008-0148-8
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/219662
dc.subjectProbability
dc.subjectRepresentation Theory
dc.subject60J60; 60C05
dc.titleInfinite-dimensional diffusions as limits of random walks on partitions
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